Relative injective modules, superstability and noetherian categories
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911990967959552 |
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| author | Mazari-Armida, Marcos Rosicky, Jiri |
| author_facet | Mazari-Armida, Marcos Rosicky, Jiri |
| contents | We study classes of modules closed under direct sums, $\mathcal{M}$-submodules and $\mathcal{M}$-epimorphic images where $\mathcal{M}$ is either the class of embeddings, $RD$-embeddings or pure embeddings.
We show that the $\mathcal{M}$-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, $RD$-injective modules, pure injective modules, flat cotorsion modules and $\mathfrak{s}$-torsion pure injective modules satisfy this criterion. The argument presented is a model theoretic one. We use in an essential way stable independence relations which generalize Shelah's non-forking to abstract elementary classes.
We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_02456 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Relative injective modules, superstability and noetherian categories Mazari-Armida, Marcos Rosicky, Jiri Rings and Algebras Logic Primary: 03C48, 13L05. Secondary: 03C45, 03C60, 16D50, 16B70, 16D10, 18G05 We study classes of modules closed under direct sums, $\mathcal{M}$-submodules and $\mathcal{M}$-epimorphic images where $\mathcal{M}$ is either the class of embeddings, $RD$-embeddings or pure embeddings. We show that the $\mathcal{M}$-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, $RD$-injective modules, pure injective modules, flat cotorsion modules and $\mathfrak{s}$-torsion pure injective modules satisfy this criterion. The argument presented is a model theoretic one. We use in an essential way stable independence relations which generalize Shelah's non-forking to abstract elementary classes. We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability. |
| title | Relative injective modules, superstability and noetherian categories |
| topic | Rings and Algebras Logic Primary: 03C48, 13L05. Secondary: 03C45, 03C60, 16D50, 16B70, 16D10, 18G05 |
| url | https://arxiv.org/abs/2308.02456 |